# Raabe criterion

A criterion for the convergence of series of complex numbers $\sum_n a_n$, proved by J. Raabe. If $a_n \neq 0$ and there is a number $R>1$ such that for sufficiently large $n$ the inequality $$\frac{|a_{n+1}|}{|a_n|} \leq 1 - \frac{R}{n}$$ holds, then $\sum_n a_n$ converges absolutely. If instead there is $N$ such that $\frac{|a_{n+1}|}{|a_n|} \geq 1 - \frac{1}{n} \qquad \forall n \geq N\, ,$ then the series $\sum_n |a_n|$ diverges, which can be easily shown comparing it to the harmonic series. However, the series itself might still converge, as can be seen taking $\sum_n (-1)^n \frac{1}{\sqrt{n}}\, .$ The number $R$ is related to the limit $\lim_{n\to \infty} n \left(1-\frac{|a_n|}{|a_{n+1}|}\right)$ and the criterion can therefore be compared to Gauss' criterion. Observe however that the harmonic series $\sum \frac{1}{n}$ (which diverges) and the series $\sum \frac{1}{n (\log n)^2}$ (which converges) have both the property that $\lim_{n\to \infty} n \left(1-\frac{a_n}{a_{n+1}}\right) = 1\, .$